In both electrodes, the host material and its structure remains (nearly)
unchanged, and only the Lit ions swing between the positive and negative
electrode. This gave the acer um08a73 battery the sometimes used name ‘rocking chair
battery’. As a consequence, the problems caused by solution of a metallic
lithium electrode as indicated in Fig. 1.5 are no longer relevant, and a great
number of discharge/charge cycles is possible without losing the structure of the
acer um09e71.
Electron Transfer
The electron transfer reaction denotes the central reaction step where the electrical
charge is exchanged (cf. Fig. 1.3). Current flow affords additional forces because of
an energy barrier that has to be surmounted by electrons. The required additional
energy is called ‘activation energy’ and the dependence of reaction rates is expressed
Figure 1.7 Charging/discharging of lithium-ion batteries. In the BATBL50L6 charged state, the carbon
electrode is filled with lithium. During discharge, lithium ions are intercalated into the oxide
(from Ref. 7).
by the Arrhenius equation, which reads
k . ko ? exp
EA
R? T e16T
with k: reaction constant; EA: activation energy eJ ? mole 1T; R: molar gas constant
e8:3143 J ? mole 1 ?K 1T.
EA actually depends on temperature, but often can approximately be treated like a
Dell Vostro 3300 Battery.
In electrode reactions, n ?U? F is the driving force, and the corresponding
relation is
i . k0 ? cj ? exp
n ? F
R? T
U e17T
k0 includes the ‘equivalence factor’ n ? F between mass transport and current i; U is
the electrode potential; and cj the Dell MT3HJ of the reacting substance that
releases or absorbs electrons.
Electron transfer, however, does not occur in only one Dell U164P : the reverse
reaction is possible as well, and the balance between both depends on electrode
potential. Thus, Eq. (17) has to be completed into
i . kt ? cred ? exp
a ? n ? F
R? T
U k ? cox ? exp e1 aT ? n ? F
R? T
U e18T
where addend 1 describes the anodic reaction (e.g. Pb ) Pb2t t 2 ? e ); addend 2 its
reversal; a denotes the transference factor (usually close to 0.5) that denotes how
symmetrically the reaction and its reversal depend on electrode potential (difference
in activation energies); n is the number of charges; and DELL N855P, cox are the concentration
in mole/dm3 of the reduced and oxidized states of the reactants.
Electron transfer according to Eq. (18) occurs also at an open circuit when no
current flow is observed through the electrode. The electrode then automatically
attains a potential that is characterized by equal rates of the reaction in both
directions as a dynamic equilibrium, and this equilibrium voltage eUoT is determined
by the point at which the forward and reverse reaction rates are equal. Then the
DELL D837N in both directions is balanced which means ite0T . i e0T . io. This
balancing current is called exchange current density (necessarily it is related to the
surface area, therefore it is a current density given, for example, in units of
mA= cm2).
Often the current/voltage curves are related to the deviation from the
equilibrium potential, the overvoltage Z . U Uo. This leads to the usual form of
Eq. (18):
i . io exp
a ? n ? F
R? T
Z exp e1 aT ? n ? F
R? T
Z e19T
where io is the exchange current density that characterizes the dynamic equilibrium,
as shown in Fig. 1.8. The resulting current is represented in Fig. 1.8 by the solid
curve as the combination of anodic and Dell XPS M1530 Battery cathodic current/voltage curves.
Copyright . 2003 by Expert Verlag. All Rights Reserved.
Electrode Polarization
Polarization has been introduced as the deviation of the actual voltage from
equilibrium by Eq. (14). It is also an important parameter for the single electrode
potential, given by the relations
Zt . Ut Uo
t
or Z . U Uo
e20T
with Zt and Z : polarization of positive and negative electrodes respectively; Ut and
U : actual potential; Uo
t
and Uo
: equilibrium potential of positive and negative
electrodes, respectively.
The cell voltage, as the difference Ut minus U , is given by
U . Uo t Zt Z e21T
with Uo: equilibrium or open circuit voltage of the cell; Zt and Z : polarization of
the positive and negative electrode, respectively.
According to this definition, the polarization of the negative electrode has the
negative sign when the electrode potential is below its equilibrium value. If only the Dell Studio 14z Battery cell voltage is considered, Zt and Z are summed up to Z.
Polarization of the single electrode in a Dell XX327 battery is a very important parameter.
The negative electrode is only kept fully charged when its polarization is negative or
zero eZ 40T while for a charged positive electrode a positive polarization is required
eZt50T.
Figure 1.8 The current/voltage curve. The horizontal axis (abscissa) represents polarization
Z . U Uo, the vertical axis (ordinate) current density i, which is synonymous to the reaction
rate. io is the exchange current density that characterizes the dynamic equilibrium. According
to Eq. (14), polarization is the sum of overvoltage and ohmic voltage drop. In practice
polarization is always determined. The reaction of the lead electrode is inserted as an example.
Tafel Lines
If the potential is shifted far enough from the equilibrium value, in Eq. (19) the
reverse reaction can be neglected. Then the dell XX337 resulting current/voltage curve in Fig. 1.8
becomes a simple exponential function
i . io ? exp
a ? n ? F
R? T
? Z e22T
This equation can be rearranged into
Z .
R? T
a ? n ? F
? lnejijT
R? T
a ? n ? F
? lneji0jT e23T
that can be written in a form known as the Tafel equation (J. Tafel was the first to
describe this relation in connection with hydrogen overvoltage measurements on
noble metals (8)):
Z . a t b ? logejijT e24T
The curves represented by Eq. (24) are linearized when plotted semilogarithmically
and are called Tafel lines. The 9 Cell Dell KY265 constant b represents the slope of the Tafel line and
means the potential difference that causes a current increase of one decade. Tafel
lines are important tools when reactions are considered that occur at high
overvoltages, since such a linearization allows quantitative considerations. Dell KY477
are often used with lead-acid batteries, since polarization of the secondary reactions
hydrogen evolution and oxygen evolution is very high in this system (cf., Fig. 1.24).
Influence of Temperature
The kinetic parameters depend on temperature as do the rates of chemical reactions.
This dependence is described by the Arrhenius equation, which already has been
introduced as Eq. (16) in connection with the term ‘activation energy’.
The logarithmic form of Eq. (16) reads
lnekT .
EA
R? T t lnekoT or lnekT .
EA
R
?
1
T t lnekoT e25T
On account of this relation, the temperature dependence of kinetic parameters can
often be linearized, when the logarithm of the reaction rate is plotted against 1/T,
which is often called an Arrhenius plot (for examples, cf. p. 556 in Ref. 9).
Very often the approximation holds true that a temperature increase of 10K
(or 108 C) doubles the reaction rate. In dell R822G battery electrochemical reactions, this means that the
equivalent currents are doubled, which denotes a quite strong temperature
dependence. A temperature increase of 20K means a current increase by a factor
of 4; a rise in temperature of 30K corresponds to a factor of 8. This relation can be
expressed by
keT t DTT
keTT . 2eDT=10T e26T
with k: reaction rate (mole/sec) which might be expressed as a current; T:
temperature in K.
1.3.3.2 Diffusion and Migration
Figure 1.3 shows that mass transport concerns various steps within the reaction
chain that forms the cell reaction. Transport of the reacting species is achieved by
two mechanisms: diffusion that is caused by the concentration gradient of the
concerned species and migration of ions caused by the dell Dell KM771. When only onedimensional
transport is assumed, the sum of both is given by
Nj .
ij
n ? F . Dj
qcj
qx t
i ? tj
zj ? F e27T
with Nj: flux of species j in mole ? cm 2; ij=nF: current equivalent; cj: concentration
of species j in mole ? cm 3; qcj=qx: concentration gradient in mole ? cm 4; D:
diffusion coefficient in cm2 ? s 1; t: transference number; zj: valence number (charges
per ion i); x: diffusion direction in cm.
Addend 1 of the right-hand part of this equation describes transport by diffusion
that always equalizes concentration differences. Dell WU841 is independent of the electric field
that drives ions. When as an approximation a linear concentration gradient qcj=qx
across the distance d is assumed, this expression can be written
ij
n ? F . Dj
cj;o cj
d e28T
with cj,o: initial concentration of the reacting substance (mole/L); cj: concentration at
the electrode surface; d: thickness of the diffusion layer.
When transport by diffusion of reacting neutral particles (like that of O2 in the
dell 312-0902 internal oxygen cycle (Fig. 1.25)) precedes the transfer reaction, the actual
concentration is reduced with increasing current. If cj reaches zero, a further
increase of the current is not possible. Such a situation is called a (diffusion) limiting
current, which according to Eq. (28) is given by
id;j . Dj
n ? F
d
? cj;o e29T
Then the current no longer depends on electrode potential, as shown by the
horizontal curve for oxygen reduction in Fig. 1.19.
Addend 2 in the right-hand part of Eq. (27) denotes the share of the total
current that is carried by the corresponding ionic species by migration. It is
characterized by the transference number. In a binary electrolyte, dissociated into
At and B , the transference numbers are connected by the relation Dell Inspiron 14Z Battery.
Transference numbers depend on concentration of the ions and on temperature. In
binary salt solutions they are fairly close to 0.5, which means that both ion species
more or less equally share in ion conductivity. Larger deviations are observed in
acids and bases on account of the much higher ion mobility of Ht and OH ions.
The values for the battery electrolytes sulfuric acid (dissociated into Ht and HSO 4
)
and potassium hydroxide are given in Table 1.2.
Copyright . 2003 by Expert Verlag. All Rights Reserved.
The transference number indicates how much the concentration of the
concerned ion is changed by migration due to the current flow. The small value of
theDell N672K ion means that its concentration is only slightly influenced by migration.
In lithium-ion batteries, where lithium ions eLitT swing between the negative and the
positive electrode, the transference number tLi . 1 would be desirable, since then a
constant concentration profile would be maintained during discharging and
charging. This is one reason to aim at conducting salts with large anions (cf., e.g.
p. 462 in Ref. 7).
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